# Properties

 Label 9075.2.a.db Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{11})$$ Defining polynomial: $$x^{4} - 7x^{2} + 4$$ x^4 - 7*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{3} + (\beta_{3} + 2) q^{4} + \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (2 \beta_{2} + \beta_1) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 + q^3 + (b3 + 2) * q^4 + b1 * q^6 + (-b2 + b1) * q^7 + (2*b2 + b1) * q^8 + q^9 $$q + \beta_1 q^{2} + q^{3} + (\beta_{3} + 2) q^{4} + \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (2 \beta_{2} + \beta_1) q^{8} + q^{9} + (\beta_{3} + 2) q^{12} + ( - 3 \beta_{2} + 2 \beta_1) q^{13} + 2 q^{14} + (\beta_{3} + 4) q^{16} + ( - 2 \beta_{2} + 4 \beta_1) q^{17} + \beta_1 q^{18} + ( - 4 \beta_{2} + \beta_1) q^{19} + ( - \beta_{2} + \beta_1) q^{21} - 8 q^{23} + (2 \beta_{2} + \beta_1) q^{24} + ( - \beta_{3} + 2) q^{26} + q^{27} + 2 \beta_{2} q^{28} + 4 \beta_1 q^{29} + \beta_{3} q^{31} + ( - 2 \beta_{2} + 3 \beta_1) q^{32} + (2 \beta_{3} + 12) q^{34} + (\beta_{3} + 2) q^{36} - 5 q^{37} + ( - 3 \beta_{3} - 4) q^{38} + ( - 3 \beta_{2} + 2 \beta_1) q^{39} + 4 \beta_{2} q^{41} + 2 q^{42} + ( - 3 \beta_{2} + 6 \beta_1) q^{43} - 8 \beta_1 q^{46} + (2 \beta_{3} + 4) q^{47} + (\beta_{3} + 4) q^{48} + ( - \beta_{3} - 4) q^{49} + ( - 2 \beta_{2} + 4 \beta_1) q^{51} + (4 \beta_{2} - 3 \beta_1) q^{52} + ( - 2 \beta_{3} + 6) q^{53} + \beta_1 q^{54} + 2 \beta_{3} q^{56} + ( - 4 \beta_{2} + \beta_1) q^{57} + (4 \beta_{3} + 16) q^{58} + 4 q^{59} + (2 \beta_{2} + \beta_1) q^{61} + (2 \beta_{2} + \beta_1) q^{62} + ( - \beta_{2} + \beta_1) q^{63} - \beta_{3} q^{64} + (3 \beta_{3} + 4) q^{67} + (8 \beta_{2} + 6 \beta_1) q^{68} - 8 q^{69} + (2 \beta_{3} + 6) q^{71} + (2 \beta_{2} + \beta_1) q^{72} - 3 \beta_{2} q^{73} - 5 \beta_1 q^{74} + (2 \beta_{2} - 9 \beta_1) q^{76} + ( - \beta_{3} + 2) q^{78} + ( - \beta_{2} - 2 \beta_1) q^{79} + q^{81} + (4 \beta_{3} + 8) q^{82} + (2 \beta_{2} + 2 \beta_1) q^{83} + 2 \beta_{2} q^{84} + (3 \beta_{3} + 18) q^{86} + 4 \beta_1 q^{87} + (4 \beta_{3} - 4) q^{89} + ( - 3 \beta_{3} + 7) q^{91} + ( - 8 \beta_{3} - 16) q^{92} + \beta_{3} q^{93} + (4 \beta_{2} + 6 \beta_1) q^{94} + ( - 2 \beta_{2} + 3 \beta_1) q^{96} + (\beta_{3} + 7) q^{97} + ( - 2 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + q^3 + (b3 + 2) * q^4 + b1 * q^6 + (-b2 + b1) * q^7 + (2*b2 + b1) * q^8 + q^9 + (b3 + 2) * q^12 + (-3*b2 + 2*b1) * q^13 + 2 * q^14 + (b3 + 4) * q^16 + (-2*b2 + 4*b1) * q^17 + b1 * q^18 + (-4*b2 + b1) * q^19 + (-b2 + b1) * q^21 - 8 * q^23 + (2*b2 + b1) * q^24 + (-b3 + 2) * q^26 + q^27 + 2*b2 * q^28 + 4*b1 * q^29 + b3 * q^31 + (-2*b2 + 3*b1) * q^32 + (2*b3 + 12) * q^34 + (b3 + 2) * q^36 - 5 * q^37 + (-3*b3 - 4) * q^38 + (-3*b2 + 2*b1) * q^39 + 4*b2 * q^41 + 2 * q^42 + (-3*b2 + 6*b1) * q^43 - 8*b1 * q^46 + (2*b3 + 4) * q^47 + (b3 + 4) * q^48 + (-b3 - 4) * q^49 + (-2*b2 + 4*b1) * q^51 + (4*b2 - 3*b1) * q^52 + (-2*b3 + 6) * q^53 + b1 * q^54 + 2*b3 * q^56 + (-4*b2 + b1) * q^57 + (4*b3 + 16) * q^58 + 4 * q^59 + (2*b2 + b1) * q^61 + (2*b2 + b1) * q^62 + (-b2 + b1) * q^63 - b3 * q^64 + (3*b3 + 4) * q^67 + (8*b2 + 6*b1) * q^68 - 8 * q^69 + (2*b3 + 6) * q^71 + (2*b2 + b1) * q^72 - 3*b2 * q^73 - 5*b1 * q^74 + (2*b2 - 9*b1) * q^76 + (-b3 + 2) * q^78 + (-b2 - 2*b1) * q^79 + q^81 + (4*b3 + 8) * q^82 + (2*b2 + 2*b1) * q^83 + 2*b2 * q^84 + (3*b3 + 18) * q^86 + 4*b1 * q^87 + (4*b3 - 4) * q^89 + (-3*b3 + 7) * q^91 + (-8*b3 - 16) * q^92 + b3 * q^93 + (4*b2 + 6*b1) * q^94 + (-2*b2 + 3*b1) * q^96 + (b3 + 7) * q^97 + (-2*b2 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 6 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 6 * q^4 + 4 * q^9 $$4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 14 q^{16} - 32 q^{23} + 10 q^{26} + 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} - 20 q^{37} - 10 q^{38} + 8 q^{42} + 12 q^{47} + 14 q^{48} - 14 q^{49} + 28 q^{53} - 4 q^{56} + 56 q^{58} + 16 q^{59} + 2 q^{64} + 10 q^{67} - 32 q^{69} + 20 q^{71} + 10 q^{78} + 4 q^{81} + 24 q^{82} + 66 q^{86} - 24 q^{89} + 34 q^{91} - 48 q^{92} - 2 q^{93} + 26 q^{97}+O(q^{100})$$ 4 * q + 4 * q^3 + 6 * q^4 + 4 * q^9 + 6 * q^12 + 8 * q^14 + 14 * q^16 - 32 * q^23 + 10 * q^26 + 4 * q^27 - 2 * q^31 + 44 * q^34 + 6 * q^36 - 20 * q^37 - 10 * q^38 + 8 * q^42 + 12 * q^47 + 14 * q^48 - 14 * q^49 + 28 * q^53 - 4 * q^56 + 56 * q^58 + 16 * q^59 + 2 * q^64 + 10 * q^67 - 32 * q^69 + 20 * q^71 + 10 * q^78 + 4 * q^81 + 24 * q^82 + 66 * q^86 - 24 * q^89 + 34 * q^91 - 48 * q^92 - 2 * q^93 + 26 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 5\nu ) / 2$$ (v^3 - 5*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4$$ b3 + 4 $$\nu^{3}$$ $$=$$ $$2\beta_{2} + 5\beta_1$$ 2*b2 + 5*b1

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.52434 −0.792287 0.792287 2.52434
−2.52434 1.00000 4.37228 0 −2.52434 −0.792287 −5.98844 1.00000 0
1.2 −0.792287 1.00000 −1.37228 0 −0.792287 −2.52434 2.67181 1.00000 0
1.3 0.792287 1.00000 −1.37228 0 0.792287 2.52434 −2.67181 1.00000 0
1.4 2.52434 1.00000 4.37228 0 2.52434 0.792287 5.98844 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9075.2.a.db yes 4
5.b even 2 1 9075.2.a.cw 4
11.b odd 2 1 inner 9075.2.a.db yes 4
55.d odd 2 1 9075.2.a.cw 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.cw 4 5.b even 2 1
9075.2.a.cw 4 55.d odd 2 1
9075.2.a.db yes 4 1.a even 1 1 trivial
9075.2.a.db yes 4 11.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{4} - 7T_{2}^{2} + 4$$ T2^4 - 7*T2^2 + 4 $$T_{7}^{4} - 7T_{7}^{2} + 4$$ T7^4 - 7*T7^2 + 4 $$T_{13}^{4} - 46T_{13}^{2} + 1$$ T13^4 - 46*T13^2 + 1 $$T_{17}^{2} - 44$$ T17^2 - 44 $$T_{19}^{4} - 79T_{19}^{2} + 1156$$ T19^4 - 79*T19^2 + 1156 $$T_{23} + 8$$ T23 + 8 $$T_{37} + 5$$ T37 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 7T^{2} + 4$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 7T^{2} + 4$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 46T^{2} + 1$$
$17$ $$(T^{2} - 44)^{2}$$
$19$ $$T^{4} - 79T^{2} + 1156$$
$23$ $$(T + 8)^{4}$$
$29$ $$T^{4} - 112T^{2} + 1024$$
$31$ $$(T^{2} + T - 8)^{2}$$
$37$ $$(T + 5)^{4}$$
$41$ $$(T^{2} - 48)^{2}$$
$43$ $$(T^{2} - 99)^{2}$$
$47$ $$(T^{2} - 6 T - 24)^{2}$$
$53$ $$(T^{2} - 14 T + 16)^{2}$$
$59$ $$(T - 4)^{4}$$
$61$ $$T^{4} - 43T^{2} + 256$$
$67$ $$(T^{2} - 5 T - 68)^{2}$$
$71$ $$(T^{2} - 10 T - 8)^{2}$$
$73$ $$(T^{2} - 27)^{2}$$
$79$ $$T^{4} - 46T^{2} + 1$$
$83$ $$T^{4} - 76T^{2} + 256$$
$89$ $$(T^{2} + 12 T - 96)^{2}$$
$97$ $$(T^{2} - 13 T + 34)^{2}$$
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